Bayesian Logistic Regression - Application in Probability Prediction of disease (Diabetes)

CapStone Project_2025

Slides: slides.html ( Go to slides.qmd to edit)

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Introduction

Diabetes mellitus (DM) is a major public health concern closely associated with factors such as obesity, age, race, and gender. Identifying these associated risk factors is essential for targeted interventions D’Angelo and Ran (2025). Logistic Regression (traditional) that estimates the association between risk factors and outcomes is insufficient in analyzing the complex healthcare data (DNA sequences, imaging, patient-reported outcomes, electronic health records (EHRs), longitudinal health measurements, diagnoses, and treatments. Zeger et al. (2020). Classical maximum likelihood estimation (MLE) yields unstable results in samples that are small, have missing data, or presents quasi- and complete separation.

Bayesian hierarchical models using Markov Chain Monte Carlo (MCMC) allow analysis of multivariate longitudinal healthcare data with repeated measures within individuals and individuals nested in a population. By integrating prior knowledge and including exogenous (e.g., age, clinical history) and endogenous (e.g., current treatment) covariates, Bayesian models provide posterior distributions and risk predictions for conditions such as pneumonia, prostate cancer, and mental disorders. Parametric assumptions remain a limitation of these models.

In Bayesian inference Chatzimichail and Hatjimihail (2023), Bayesian inference has shown that parametric models (with fixed parameters) often underperform compared to nonparametric models, which do not assume a prior distribution. Posterior probabilities from Bayesian approaches improve disease classification and better capture heterogeneity in skewed, bimodal, or multimodal data distributions. Bayesian nonparametric models are flexible and robust, integrating multiple diagnostic tests and priors to enhance accuracy and precision, though reliance on prior information and restricted access to resources can limit applicability. Combining Bayesian methods with other statistical or computational approaches helps address systemic biases, incomplete data, and non-representative datasets.

The Bayesian framework described by R. van de Schoot et al. (2021) highlights the role of priors, data modeling, inference, posterior sampling, variation inference, and variable selection. Proper variable selection mitigates multicollinearity, overfitting, and limited sampling, improving predictive performance. Priors can be informative, weakly informative, or diffuse, and can be elicited from expert opinion, generic knowledge, or data-driven methods. Sensitivity analysis evaluates the alignment of priors with likelihoods, while MCMC simulations (e.g., brms, blavaan in R) empirically estimate posterior distributions. Spatial and temporal Bayesian models have applications in large-scale cancer genomics, identifying molecular interactions, mutational signatures, patient stratification, and cancer evolution, though temporal autocorrelation and subjective prior elicitation can be limiting.

Bayesian normal linear regression has been applied in metrology for instrument calibration using conjugate Normal–Inverse-Gamma priors Klauenberg et al. (2015). Hierarchical priors add flexibility by modeling uncertainty across multiple levels, improving robustness and interpretability. Bayesian hierarchical/meta-analytic linear regression incorporates both exchangeable and unexchangeable prior information, addressing multiple testing challenges, small sample sizes, and complex relationships among regression parameters across studies Leeuw and Klugkist (2012)

A sequential clinical reasoning model Liu et al. (2013) Sequential clinical reasoning models demonstrate screening by adding predictors stepwise: (1) demographics, (2) metabolic components, and (3) conventional risk factors, incorporating priors and mimicking clinical evaluation. This approach captures ecological heterogeneity and improves baseline risk estimation, though interactions between predictors and external cross-validation remain limitations.

Bayesian multiple imputation with logistic regression addresses missing data in clinical research Austin et al. (2021) in clinical research by classifying missing values (e.g., patient refusal, loss to follow-up, mechanical errors) as MAR, MNAR, or MCAR. Multiple imputation generates plausible values across datasets and pools results for reliable classification of patient health status and mortality.

Aims

This study aims to apply Bayesian logistic regression to predict diabetes status and examine its association with body mass index (BMI), age (≥20 years), gender, and race using data from the 2013–2014 NHANES survey. The NHANES dataset employs a complex sampling design involving stratification, clustering, and oversampling of specific population subgroups. By adopting a Bayesian analytical framework, this study seeks to overcome challenges commonly encountered in traditional logistic regression—such as missing data, separation, and biases introduced by complete case analysis—thereby improving the efficiency, robustness, and interpretability of model estimates in predicting population-level health outcomes.

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Method

Bayesian Logistic Regression

To estimate the associations between predictors and outcome probabilities. Bayesian framework integrates prior information with observed data to generate posterior distributions, allowing direct probabilistic interpretation of parameters.
This approach provides flexibility in model specification, accounts for parameter uncertainty, and produces credible intervals that fully reflect uncertainty in the estimates.
Unlike traditional frequentist methods, the Bayesian method enables inference through posterior probabilities, supporting more nuanced decision-making and interpretation.

Model Structure

• Bayesian logistic regression is a probabilistic modeling framework used to estimate the relationship between one or more predictors (continuous or categorical) and a binary outcome (e.g., presence/absence of disease).
  

• It extends classical logistic regression by combining it with Bayesian inference, treating model parameters as random variables with probability distributions rather than fixed point estimates Leeuw and Klugkist (2012) and Klauenberg et al. (2015)

• Bayesian logistic regression models the log-odds of a binary outcome as a linear combination of predictors:

• In the Bayesian framework, model parameters ( ) are treated as random variables and assigned prior distributions that reflect existing knowledge or plausible ranges before observing the data. After incorporating the observed evidence, the priors are updated through Bayes’ theorem (Leeuw and Klugkist 2012; Klauenberg et al. 2015):

• In the Bayesian framework, the coefficients () are assigned prior distributions, which are updated in light of the observed data to yield posterior distributions.

• The Bayesian approach naturally incorporates uncertainty in all model parameters.
  

• It combines prior beliefs with observed data to produce posterior distributions according to Bayes’ theorem: [ ]

  • Likelihood: is the probability of the observed data given the model parameters (as in classical logistic regression).
    
  • Prior: Encodes prior knowledge or beliefs about parameter values before observing the data.
    
  • Posterior: is the updated beliefs about parameters after observing the data.
  • Posterior ⋉ Likelihood * Prior

Prior Specification - Regression coefficients prior: normal(0, 2.5) — providing weakly informative regularization provide gentle regularization, constraining extreme values without overpowering the data R. van de Schoot et al. (2021) - Intercept prior: Student’s t-distribution prior, student_t(3, 0, 10) (van de Schoot et al., 2013).
This weakly informative prior:
- Has 3 degrees of freedom** (( = 3 )), producing heavy tails that allow for occasional large effects.
- Is centered at 0 (( = 0 )), reflecting no initial bias toward positive or negative associations.
- Has a scale parameter of 10 (( = 10 )), allowing broad variation in possible coefficient values.
Such priors improve stability in models with small sample sizes, high collinearity, or potential outliers.

Advantages of Bayesian Logistic Regression

• Uncertainty quantification: Produces full posterior distributions instead of single-point estimates.
• Credible intervals: Provide the range within which a parameter lies with a specified probability (e.g., 95%).
• Flexible priors: Allow incorporation of expert knowledge or results from previous studies.
• Probabilistic predictions: Posterior predictive distributions yield direct probabilities for future observations.
• Comprehensive model checking: Posterior predictive checks (PPCs) evaluate how well simulated outcomes reproduce observed data.

Posterior Predictions

• Posterior distributions of the coefficients are used to estimate the probability of the outcome for given predictor values. This enables statements such as:
   “Given the predictors, the probability of the outcome lies between X% and Y%.”
• Posterior predictions incorporate two sources of uncertainty:
  • Parameter uncertainty: Variability in estimated model coefficients.
  • Predictive uncertainty: Variability in future outcomes given those parameters.
• In Bayesian analysis, every unknown parameter — such as a regression coefficient, mean, or variance — is treated as a random variable with a probability distribution that expresses uncertainty given the observed data.

Model Evaluation and Diagnostics

• Model quality and convergence are assessed using standard Bayesian diagnostics:
• Posterior inference performed using Markov Chain Monte Carlo (MCMC) sampling via the No-U-Turn Sampler (NUTS), a variant of Hamiltonian Monte Carlo (HMC). Four chains run with sufficient warm-up iterations to ensure convergence. Austin et al. (2021)
• Convergence diagnostics: Markov Chain Monte Carlo (MCMC) performance was evaluated using ( ) (R-hat) and effective sample size (ESS).
• Autocorrelation checks: Ensure independence between successive MCMC draws.
• Posterior predictive checks (PPC): Compare simulated data from posterior distributions to observed outcomes.
• Bayesian R²: Quantified the proportion of variance explained by the predictors, incorporating uncertainty.

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Analysis and Results

Bayesian logistic regression analysis is performed to estimate the risk of doctor-diagnosed diabetes among adults in the National Health and Nutrition Examination Survey (NHANES, 2013-2014), a 2-year data cross-sectional weighted data from the CDC’s National Center for Health Statistics (National Center for Health Statistics (NCHS) 2014 Center for Health Statistics (1999).

Data pre-processing

Data is imported using R packages and libraries for data wrangling and analysis. All computation is implemented in a probabilistic programming environment using the brms interface to Stan.

Three NHANES component files—Demographics (DEMO_H), Body Measures (BMX_H), and the Diabetes Questionnaire (DIQ_H) imported in .xpt format are merged using the unique participant identifier (SEQN) to create a single analytic dataset. Prior to merge, all variables of interest were harmonized and recoded to ensure consistent numeric and factor data types, resulting in a clean, atomic structure suitable for survey-weighted analyses and Bayesian modeling.

The merged dataset with 10,175 participants was filtered to include only adults (age ≥ 20) for subsequent analyses including demographic and anthropometric measures to provide a representative sample for assessing diabetes risk across the U.S. adult population reflecting the complex, multistage sampling design of NHANES.

Variable Descriptions: Adult Dataset

Variable	Description	Type
diabetes_dx	Diabetes diagnosis (1 = Yes, 0 = No) based on medical questionnaire.	Categorical
age	Age of participant in years.	Continuous
bmi	Body Mass Index (BMI) in kilograms per square meter (kg/m²), calculated from measured height and weight.	Continuous
sex	Sex of participant (Male or Female).	Categorical
race	Race/Ethnicity (e.g., Non-Hispanic White, Non-Hispanic Black, Mexican American, etc.).	Categorical
WTMEC2YR	Examination sample weight for MEC (Mobile Examination Center) participants.	Weight
SDMVPSU	Primary Sampling Unit (PSU) used for variance estimation in complex survey design.	Design
SDMVSTRA	Stratum variable used to define strata for complex survey design.	Design
age_c	Age variable centered and standardized (z-score).	Continuous
bmi_c	BMI variable centered and standardized (z-score).	Continuous
wt_norm	Normalized survey weight (WTMEC2YR divided by its mean, for model weighting).	Weight

Variable Description

• Variables of Interest
• Response Variable - Binary Type 2 / diagnosed diabetes(excluding gestational diabetes) diabetes_dx created combning - DIQ010 - Doctor told you have diabetes - DIQ050- excluded (a secondary variable describing treatment status (insulin use)).
• Predictor Variables - Body Mass Index, factor, 4 levels
  
• Covariates - Gender (factor, 2 levels) - Ethnicity (factor, 5 levels) - Age (continuous 20-80 years)
• Survey design variables — primary sampling units (SDMVPSU), strata (SDMVSTRA), and examination weights (WTMEC2YR) — were retained to account for this design to account fpr unequal probabilities of selection and nonresponse and produce population-representative estimates, valid standard errors and inference for U.S. adults.

TABLE

Step	Description- recoding, categorization, missing data and final data
Weighting	Used the survey package to calculate weighted means and standard deviations for all variables.
Standardization	Standardized BMI and age variables for analysis.
Age Categorization	Recoded into intervals: 20–<30, 30–<40, 40–<50, 50–<60, 60–<70, and 70–80 years.
BMI Categorization	Recoded and categorized as: <18.5 (Underweight), 18.5–<25 (Normal), 25–<30 (Overweight), 30–<35 (Obesity I), 35–<40 (Obesity II), ≥40 (Obesity III).
Ethnicity Recoding	Recoded as: 1 = Mexican American, 2 = Other Hispanic, 3 = Non-Hispanic White, 4 = Non-Hispanic Black, 5 = Other/Multi.
Special Codes	Special codes (e.g., 3, 7) were transformed to NA. These codes are not random and could introduce bias if ignored (MAR or MNAR).
Missing Data	Missing values were retained and visualized to assess their pattern and informativeness.
Final Dataset	Created a cleaned analytic dataset (adult) using Non-Hispanic White and Male as reference groups for analysis.

      mean     SE
age 47.496 0.3805

                mean     SE
diabetes_dx 0.089016 0.0048

            variance     SE
diabetes_dx   4759.9 0.0039

Effective sample size for diabetes_dx: 48142 

Exploratory Data Analysis (Adult, 20 - 80 years)

library(dplyr)
library(skimr)
library(knitr)
library(tidyr)
library(purrr)
library(forcats)
library(kableExtra)

str(adult)

'data.frame':   5769 obs. of  12 variables:
 $ SDMVPSU    : num  1 1 1 2 1 1 2 1 2 2 ...
 $ SDMVSTRA   : num  112 108 109 116 111 114 106 112 112 113 ...
 $ WTMEC2YR   : num  13481 24472 57193 65542 25345 ...
 $ diabetes_dx: num  1 1 1 0 0 0 0 0 0 0 ...
 $ bmi        : num  26.7 28.6 28.9 19.7 41.7 35.7 NA 26.5 22 20.3 ...
 $ age        : num  69 54 72 73 56 61 42 56 65 26 ...
 $ sex        : Factor w/ 2 levels "Male","Female": 1 1 1 2 1 2 1 2 1 2 ...
 $ race       : Factor w/ 5 levels "NH White","Mexican American",..: 4 1 1 1 2 1 3 1 1 1 ...
 $ DIQ050     : num  1 1 1 2 2 2 2 2 2 2 ...
 $ age_c      : num  1.132 0.278 1.303 1.36 0.392 ...
 $ bmi_c      : num  -0.3359 -0.0703 -0.0283 -1.3144 1.761 ...
 $ bmi_cat    : Factor w/ 6 levels "<18.5","18.5–<25",..: 3 3 3 2 6 5 NA 3 2 2 ...

plot_intro(adult, title="Figure 1 (Adult dataset). Structure of variables and missing observations.")

plot_missing(adult, title="Figure 2(Adult dataset). Breakdown of missing observations.")

Missingness

• 25% of columns are discrete (categorical), 75% are continuous (age and BMI).
• 92.7% of rows have complete information for all variables with fully observed data across predictors and outcomes for most participants.
• Only 1.3% of individual data points are missing across the dataset, reflecting minimal missingness.
• No column is entirely missing (0%), indicating all variables have at least some observed data.
• Overall missingness: ~4% → low, but non-trivial given the small number of variables involved.

Study Cohort

• The adult cohort includes standardized variables for age and BMI (age_c, bmi_c), categorical indicators for sex and race/ethnicity, and a binary doctor-diagnosed diabetes.
• The adult cohort included 5,769 adults aged 20–80 years with the mean age of participants was 49.1 years (SD = 17.6), and participants are fairly evenly distributed across adult age groups, with no sharp skewness.
• The mean BMI was 29.1 kg/m² (SD = 7.15), with values ranging from 14.1 to 82.9 kg/m².A small proportion of missing values in variables (BMI and diabetes status). BMI data were available for 5,520 participants, with 249 missing values. BMI: Most participants have BMI values within the normal to overweight range, with fewer in the obese category.
• Sex: Sample includes a higher proportion of females than males.

ggplot(adult, aes(x = age)) +
  geom_histogram(binwidth = 5, fill = "skyblue", color = "white") +
  labs(
    title = "Distribution of Age >20 years",
    x = "Age (years)",
    y = "Count"
  ) +
  theme_minimal()

ggplot(adult, aes(factor(diabetes_dx))) +
  geom_bar(fill = "steelblue") +
  labs(title="Diabetes Outcome Distribution in >20 years age group", x="diabetes_dx (0=No, 1=Yes)", y="Count")

ggplot(adult, aes(factor(bmi_cat))) +
  geom_bar(fill = "steelblue") +
  labs(title="Diabetes Outcome Distribution by BMI in >20 years age group", x="bmi_cat")

ggplot(adult, aes(x = factor(diabetes_dx), y = bmi)) +
  geom_boxplot(fill = "skyblue") +
  labs(
    title = "BMI Distribution by Diabetes Diagnosis in >20 years age group",
    x = "Diabetes Diagnosis (0 = No, 1 = Yes)",
    y = "BMI"
  ) +
  theme_minimal()

# plots for adult data bmi categories and race categories

ggplot(adult, aes(x = factor(race), fill = factor(diabetes_dx))) +
  geom_bar(position = "dodge") +
  labs(
    title = "Diabetes Diagnosis by Race in >20 years age group",
    x = "Race/Ethnicity",
    y = "Count",
    fill = "Diabetes Diagnosis\n(0 = No, 1 = Yes)"
  ) +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1))

ggplot(adult, aes(x = factor(bmi_cat), fill = factor(diabetes_dx))) +
  geom_bar(position = "dodge") +
  labs(
    title = "Diabetes Diagnosis by BMI in >20 years age group",
    x = "BMI",
    y = "Count",
    fill = "Diabetes Diagnosis\n(0 = No, 1 = Yes)"
  ) +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1))

Prevalence of diabetes in adult population

• 1. Diabetes by BMI categories: Individuals diagnosed with diabetes tend to have higher BMI values compared to non-diabetics.
• 2. Diabetesby Age Group: The proportion of diabetes increases with advancing age, highlighting age as a strong risk factor.
• 3. Diabetes by Race/Ethnicity: Differences are observed across racial/ethnic groups, with some showing higher prevalence rates than others.
  4. Diabetes diagnosis by sex across different racial groups. Bars are side by side for each sex, with counts displayed on top

# Example: create your dataset
adult1 <- data.frame(
  race = rep(c("NH White","Mexican American","Other Hispanic","NH Black","Other/Multi"), each = 6),
  sex = rep(c("Male","Male","Male","Female","Female","Female"), times = 5),
  diabetes_dx = rep(c(0,1,NA,0,1,NA), times = 5),
  count = c(
    1019,119,38,1164,96,36,
    304,60,14,329,49,11,
    183,26,10,255,25,9,
    461,100,19,515,65,17,
    351,46,8,393,32,15
  )
)

# Clean NA for plotting or convert to "Missing"
adult1$diabetes_dx <- as.character(adult1$diabetes_dx)
adult1$diabetes_dx[is.na(adult1$diabetes_dx)] <- "Missing"

# Plot grouped bar chart
ggplot(adult1, aes(x = diabetes_dx, y = count, fill = sex)) +
  geom_bar(stat = "identity", position = "dodge") +
  facet_wrap(~race) +
  labs(title = "Diabetes Diagnosis by Sex and Race",
       x = "Diabetes Diagnosis",
       y = "Count") +
  theme_minimal() +
  scale_fill_manual(values = c("skyblue", "orange"))

Statistical Modeling

Analytic workflow: Three modeling frameworks were compared using identical predictors (standardized age, BMI, sex, and race4) and the binary outcome diabetes_dx: - (1) survey-weighted logistic regression to incorporate the NHANES complex sampling design,
- (2) multiple imputation (MICE) to address missing BMI values, and - - (3) Bayesian logistic regression with weakly informative priors to quantify uncertainty.

# Modeling

library(broom)
library(mice)
library(brms)
library(posterior)
library(bayesplot)
library(knitr)

# --- Guardrails for modeling ---
n_outcome <- sum(!is.na(adult$diabetes_dx))
if (n_outcome == 0) stop("Too few non-missing outcomes for modeling. n = 0")

# Ensure factors and >=2 observed levels among complete outcomes
adult <- adult %>%
  dplyr::mutate(
    sex  = if (!is.factor(sex))  factor(sex)  else sex,
    race = if (!is.factor(race)) factor(race) else race
  )

if (nlevels(droplevels(adult$sex[!is.na(adult$diabetes_dx)]))  < 2)
  stop("sex has <2 observed levels after filtering; check data availability.")
if (nlevels(droplevels(adult$race[!is.na(adult$diabetes_dx)])) < 2)
  stop("race has <2 observed levels after filtering; check Data Prep.")

   #  Survey-weighted complete-case 
# Build a logical filter on the original adult data (same length as design$data)
keep_cc <- with(
  adult,
  !is.na(diabetes_dx) & !is.na(age_c) & !is.na(bmi_c) &
  !is.na(sex) & !is.na(race)
)

# Subset the survey design using the logical vector (same length as original)
des_cc <- subset(nhanes_design_adult, keep_cc)

# Corresponding complete-case data (optional)
cc <- adult[keep_cc, ] |> droplevels()
cat("\nComplete-case N for survey-weighted model:", nrow(cc), "\n")

Complete-case N for survey-weighted model: 5349 

print(table(cc$race))

        NH White Mexican American   Other Hispanic         NH Black 
            2293              713              470             1101 
     Other/Multi 
             772 

print(table(cc$diabetes_dx))

   0    1 
4752  597 

print(table(cc$sex))

  Male Female 
  2551   2798 

form_cc <- diabetes_dx ~ age_c + bmi_c + sex + race
svy_fit <- survey::svyglm(formula = form_cc, design = des_cc, family = quasibinomial())
summary(svy_fit)

Call:
svyglm(formula = form_cc, design = des_cc, family = quasibinomial())

Survey design:
subset(nhanes_design_adult, keep_cc)

Coefficients:
                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)          -2.67143    0.11935 -22.383 1.68e-08 ***
age_c                 1.10833    0.05042  21.981 1.94e-08 ***
bmi_c                 0.63412    0.05713  11.099 3.88e-06 ***
sexFemale            -0.63844    0.10926  -5.843 0.000386 ***
raceMexican American  0.71091    0.13681   5.196 0.000826 ***
raceOther Hispanic    0.46469    0.13474   3.449 0.008712 ** 
raceNH Black          0.51221    0.15754   3.251 0.011677 *  
raceOther/Multi       0.84460    0.17756   4.757 0.001433 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for quasibinomial family taken to be 0.8455444)

Number of Fisher Scoring iterations: 6

Multiple Logistic Regression model (Survey weighted Modeling) and complete-case Analysis

• Strongest predictors: Age and BMI show the largest ORs among continuous variables—both strongly linked to diabetes risk.
• Protective factor: Being female reduces the odds of diabetes.
• Race disparities: All racial/ethnic minority groups show significantly higher odds compared to Non-Hispanic Whites, consistent with known health disparities in diabetes prevalence.
• Significance: All p-values < 0.05, so all predictors are statistically significant.

Results from Survey-weighted logistic regression

• Regression coefficients for predictors of diabetes diagnosis (diabetes_dx) with the reference group (Male, Non-Hispanic White, average BMI and age) represents the change in log-odds of diabetes associated with a one-unit increase in the predictor (or compared to the reference group), controlling for all other variables.

• Baseline log-odds of diabetes = -2.69

• For each 1 SD increase in age, the log-odds of diabetes increase by 1.07 → odds increase by exp(1.07) = 2.9× (≈3× higher odds).

• For each 1 SD increase in BMI, odds of diabetes increase by exp(0.55) = 1.73× (≈73% higher).

• Females have exp(-0.62) = 0.54× the odds of diabetes compared to males → about 46% lower odds.

• Mexican Americans have exp(0.89) = 2.43× higher odds of diabetes vs. Non-Hispanic Whites. Other Hispanics have exp(0.56) = 1.75× higher odds Non-Hispanic Blacks have exp(0.68) = 1.97× higher odds. Those identifying as “Other/Multi-racial” have exp(0.75) = 2.12× higher odds of diabetes.

Interpretation - Age and BMI are strong positive predictors of diabetes - Females show significantly lower odds compared to males. - Race/Ethnicity: All non-White racial groups have significantly higher odds of diabetes, highlighting persistent disparities in diabetes risk. - All predictors are statistically significant (p < 0.01), suggesting a robust association across demographic and health variables.

svy_or <- broom::tidy(svy_fit, conf.int = TRUE) %>%
  dplyr::mutate(OR = exp(estimate), LCL = exp(conf.low), UCL = exp(conf.high)) %>%
  dplyr::select(term, OR, LCL, UCL, p.value) %>%
  dplyr::filter(term != "(Intercept)")
knitr::kable(svy_or, caption = "Survey-weighted odds ratios (per 1 SD)")

Survey-weighted odds ratios (per 1 SD)

term	OR	LCL	UCL	p.value
age_c	3.0292807	2.6967690	3.4027912	0.0000000
bmi_c	1.8853571	1.6526296	2.1508579	0.0000039
sexFemale	0.5281132	0.4104905	0.6794397	0.0003857
raceMexican American	2.0358434	1.4850041	2.7910081	0.0008262
raceOther Hispanic	1.5915182	1.1664529	2.1714810	0.0087119
raceNH Black	1.6689718	1.1605895	2.4000450	0.0116773
raceOther/Multi	2.3270527	1.5451752	3.5045697	0.0014331

Multivariate Imputation by Chained Equations (Pooled Logistic Regression)

MICE, as an alternative to the Bayesian Approach Buuren and Groothuis-Oudshoorn (2011), manages missiging data.

• Flatness of the density, heavy tails, non-zero peakedness, skewness and multimodality do not hamper performance of multiple imputation for the mean structure (n > 400) even for high percentages (75%) of missing data in one variable Van Buuren and Van Buuren (2012).
• Multiple Imputation (MI) performed using mice R package, imputes missing values of one variable at a time, using regression models based on the other variables in the dataset.
• In the chain process, each imputed variable become a predictor for the subsequent imputation, and the entire process is repeated multiple times to create several complete datasets, each reflecting different possibilities for the missing data.

# ----- Multiple Imputation (predictors only) 
mi_dat <- adult %>%
  dplyr::select(diabetes_dx, age, bmi, sex, race, WTMEC2YR, SDMVPSU, SDMVSTRA)

meth <- mice::make.method(mi_dat)
pred <- mice::make.predictorMatrix(mi_dat)

# Do not impute outcome
meth["diabetes_dx"] <- ""
pred["diabetes_dx", ] <- 0
pred[,"diabetes_dx"] <- 1

# Imputation methods
meth["age"]  <- "norm"
meth["bmi"]  <- "pmm"
meth["sex"]  <- "polyreg"
meth["race"] <- "polyreg"

# Survey design vars as auxiliaries only
meth[c("WTMEC2YR","SDMVPSU","SDMVSTRA")] <- ""
pred[, c("WTMEC2YR","SDMVPSU","SDMVSTRA")] <- 1

glimpse(mi_dat)

Rows: 5,769
Columns: 8
$ diabetes_dx <dbl> 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ age         <dbl> 69, 54, 72, 73, 56, 61, 42, 56, 65, 26, 76, 33, 32, 38, 50…
$ bmi         <dbl> 26.7, 28.6, 28.9, 19.7, 41.7, 35.7, NA, 26.5, 22.0, 20.3, …
$ sex         <fct> Male, Male, Male, Female, Male, Female, Male, Female, Male…
$ race        <fct> NH Black, NH White, NH White, NH White, Mexican American, …
$ WTMEC2YR    <dbl> 13481.04, 24471.77, 57193.29, 65541.87, 25344.99, 61758.65…
$ SDMVPSU     <dbl> 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2…
$ SDMVSTRA    <dbl> 112, 108, 109, 116, 111, 114, 106, 112, 112, 113, 116, 114…

imp <- mice::mice(mi_dat, m = 5, method = meth, predictorMatrix = pred, seed = 123)

 iter imp variable
  1   1  bmi
  1   2  bmi
  1   3  bmi
  1   4  bmi
  1   5  bmi
  2   1  bmi
  2   2  bmi
  2   3  bmi
  2   4  bmi
  2   5  bmi
  3   1  bmi
  3   2  bmi
  3   3  bmi
  3   4  bmi
  3   5  bmi
  4   1  bmi
  4   2  bmi
  4   3  bmi
  4   4  bmi
  4   5  bmi
  5   1  bmi
  5   2  bmi
  5   3  bmi
  5   4  bmi
  5   5  bmi

Results from MICE:

• After MICE, the final pooled imputed dataset consisted of 5,769 participants with 8 variables with missing values were addressed
• Five imputations across five iterations each, with BMI imputed conditionally based on other predictors (age, sex, race, and diabetes status).
• The iterative process showed stable convergence, indicating reliable estimation of missing BMI values for subsequent survey-weighted and Bayesian modeling analyses.

fit_mi <- with(imp, {
  age_c <- as.numeric(scale(age))
  bmi_c <- as.numeric(scale(bmi))
  glm(diabetes_dx ~ age_c + bmi_c + sex + race, family = binomial())
})
pool_mi <- pool(fit_mi)
summary(pool_mi)

                  term   estimate  std.error  statistic       df       p.value
1          (Intercept) -2.6895645 0.09941301 -27.054453 5566.204 1.486581e-151
2                age_c  1.0660265 0.05594733  19.054108 5520.446  1.911564e-78
3                bmi_c  0.5468538 0.04473386  12.224604 5148.557  6.751227e-34
4            sexFemale -0.6178297 0.09379129  -6.587282 5551.660  4.892566e-11
5 raceMexican American  0.8877355 0.13750463   6.456041 5472.583  1.167455e-10
6   raceOther Hispanic  0.5606621 0.17485537   3.206433 5573.987  1.351505e-03
7         raceNH Black  0.6809629 0.11981185   5.683602 5576.734  1.385727e-08
8      raceOther/Multi  0.7476406 0.15300663   4.886328 4749.963  1.061140e-06

## table 

mi_or <- summary(pool_mi, conf.int = TRUE, exponentiate = TRUE) %>%
  dplyr::rename(
    term = term, OR = estimate, LCL = `2.5 %`, UCL = `97.5 %`, p.value = p.value
  ) %>%
  dplyr::filter(term != "(Intercept)")
knitr::kable(mi_or, caption = "MI pooled odds ratios (per 1 SD)")

MI pooled odds ratios (per 1 SD)

	term	OR	std.error	statistic	df	p.value	LCL	UCL	conf.low	conf.high
2	age_c	2.9038183	0.0559473	19.054108	5520.446	0.0000000	2.6021752	3.2404277	2.6021752	3.2404277
3	bmi_c	1.7278084	0.0447339	12.224604	5148.557	0.0000000	1.5827382	1.8861754	1.5827382	1.8861754
4	sexFemale	0.5391132	0.0937913	-6.587282	5551.660	0.0000000	0.4485669	0.6479368	0.4485669	0.6479368
5	raceMexican American	2.4296216	0.1375046	6.456041	5472.583	0.0000000	1.8555327	3.1813298	1.8555327	3.1813298
6	raceOther Hispanic	1.7518320	0.1748554	3.206433	5573.987	0.0013515	1.2434346	2.4680953	1.2434346	2.4680953
7	raceNH Black	1.9757793	0.1198118	5.683602	5576.734	0.0000000	1.5621842	2.4988753	1.5621842	2.4988753
8	raceOther/Multi	2.1120110	0.1530066	4.886328	4749.963	0.0000011	1.5646727	2.8508138	1.5646727	2.8508138

Descriptive Statistics of Imputed dataset - age (mean (SD)) = 48.84 (17.57) - females 2923 (52.3%) > males 2669 (47.7%) - majority being NH White count = 2398 (42.9%) - non-diabetics: Diabetics :: 4974 (88.9%):618 (11.1%)

library(gt)

# Bayesian Logistic Regression (formula weights) 
adult_imp1 <- complete(imp, 1) %>%
  dplyr::mutate(
    age_c  = as.numeric(scale(age)),
    bmi_c  = as.numeric(scale(bmi)),
    wt_norm = WTMEC2YR / mean(WTMEC2YR, na.rm = TRUE),
    # ensure factor refs match survey/mice:
    race = forcats::fct_relevel(race, "NH White"),
    sex  = forcats::fct_relevel(sex,  "Male")
  ) %>%
  dplyr::filter(!is.na(diabetes_dx), !is.na(age_c), !is.na(bmi_c),
                !is.na(sex), !is.na(race)) %>%
  droplevels()

stopifnot(all(is.finite(adult_imp1$wt_norm)))

glimpse(adult_imp1)

Rows: 5,592
Columns: 11
$ diabetes_dx <dbl> 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ age         <dbl> 69, 54, 72, 73, 56, 61, 42, 56, 65, 26, 76, 33, 32, 38, 50…
$ bmi         <dbl> 26.7, 28.6, 28.9, 19.7, 41.7, 35.7, 23.6, 26.5, 22.0, 20.3…
$ sex         <fct> Male, Male, Male, Female, Male, Female, Male, Female, Male…
$ race        <fct> NH Black, NH White, NH White, NH White, Mexican American, …
$ WTMEC2YR    <dbl> 13481.04, 24471.77, 57193.29, 65541.87, 25344.99, 61758.65…
$ SDMVPSU     <dbl> 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2…
$ SDMVSTRA    <dbl> 112, 108, 109, 116, 111, 114, 106, 112, 112, 113, 116, 114…
$ age_c       <dbl> 1.13241831, 0.27835981, 1.30323001, 1.36016725, 0.39223428…
$ bmi_c       <dbl> -0.33319172, -0.06755778, -0.02561558, -1.31184309, 1.7639…
$ wt_norm     <dbl> 0.3393916, 0.6160884, 1.4398681, 1.6500477, 0.6380722, 1.5…

library(tableone)

vars <- c("age", "bmi", "age_c", "bmi_c", "wt_norm", "sex", "race", "diabetes_dx")

table1 <- CreateTableOne(vars = vars, data = adult_imp1, factorVars = c("sex", "race", "diabetes_dx"))
print(table1, showAllLevels = TRUE)

                     
                      level            Overall      
  n                                     5592        
  age (mean (SD))                      48.84 (17.57)
  bmi (mean (SD))                      29.00 (7.11) 
  age_c (mean (SD))                    -0.02 (1.00) 
  bmi_c (mean (SD))                    -0.01 (0.99) 
  wt_norm (mean (SD))                   1.00 (0.79) 
  sex (%)             Male              2669 (47.7) 
                      Female            2923 (52.3) 
  race (%)            NH White          2398 (42.9) 
                      Mexican American   742 (13.3) 
                      Other Hispanic     489 ( 8.7) 
                      NH Black          1141 (20.4) 
                      Other/Multi        822 (14.7) 
  diabetes_dx (%)     0                 4974 (88.9) 
                      1                  618 (11.1) 

## correlation matrix
library(ggplot2)
library(reshape2)

correlation_matrix <- cor(adult_imp1[, c("diabetes_dx", "age", "bmi")], use = "complete.obs", method = "pearson")
correlation_melted <- melt(correlation_matrix)

ggplot(correlation_melted, aes(Var1, Var2, fill = value)) +
  geom_tile(color = "white") +
  scale_fill_gradient2(low = "blue", high = "red", mid = "white", midpoint = 0,
                       limit = c(-1, 1), space = "Lab", name = "Correlation") +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1)) +
  labs(title = "Correlation Heatmap", x = "Features", y = "Features")

Visualization of the imputed dataset: (1) Correlation matrix: Pairwise correlations heatmap: show the strength and direction of correlations (Pearson correlation) which measures linear association between diabetes_dx, age, and bmi (2) Diabetes Diagnosis Distribution (3) BMI Distribution by Diabetes Status (4) Predicted Probability of Diabetes vs BMI

# Class distribution

ggplot(adult_imp1, aes(x = factor(diabetes_dx))) +
  geom_bar(fill = "steelblue") +
  labs(
    title = "Diabetes Diagnosis Distribution",
    x = "Diabetes Diagnosis (0 = No, 1 = Yes)",
    y = "Count"
  ) +
  theme_minimal()

prop.table(table(adult_imp1$diabetes_dx))

       0        1 
0.889485 0.110515 

# Visualization of Diabetes vs BMI (adult_data1)

library(ggplot2)

# Create the plot
ggplot(adult_imp1, aes(x = factor(diabetes_dx), y = bmi, fill = factor(diabetes_dx))) +
  geom_boxplot(alpha = 0.7) +
  scale_x_discrete(labels = c("0" = "No Diabetes", "1" = "Diabetes")) +
  labs(
    x = "Diabetes Diagnosis",
    y = "BMI",
    title = "BMI Distribution by Diabetes Status"
  ) +
  theme_minimal() +
  theme(legend.position = "none")

# logistic regression curve
ggplot(adult_imp1, aes(x = bmi, y = diabetes_dx)) +
  geom_point(aes(y = diabetes_dx), alpha = 0.2, position = position_jitter(height = 0.02)) +
  geom_smooth(method = "glm", method.args = list(family = "binomial"), se = TRUE, color = "blue") +
  labs(
    x = "BMI",
    y = "Probability of Diabetes",
    title = "Predicted Probability of Diabetes vs BMI"
  ) +
  theme_minimal()

Study population accross the three datasets

• Rows: 10175 and Columns: 10 (survey-weighted, merged data)
• Rows: 5,769 and Columns: 12 (filtered data, adult)
• Rows: 5,592 and Columns: 11 (imputed data, adult_imp1)

Bayesian Logistic Regression and Analysis

1. A Bayesian logistic regression model fitted on the first imputed dataset (adult_imp1) - with survey weights-Normalized MEC exam weights (wt_norm) with mean 1.00 (SD 0.79) with no missing values and assessed predictors of diabetes diagnosis - with continuous variables standardized, categorical variables correctly re-leveled for reference categories
2. Prior Specification for the study model
   • Intercept prior: student_t(3, 0, 10) — allowing heavy tails for flexibility in the intercept estimate. R. V. D. Schoot et al. (2013)
   • Regression coefficients prior: normal(0, 2.5) — providing weakly informative regularization provide gentle regularization, constraining extreme values without overpowering the data R. van de Schoot et al. (2021)
3. Model Estimation
   • Four Markov Chain Monte Carlo (MCMC) chains, each with 2000 iterations (50% warm-up), and an adaptive delta of 0.95 ensured good chain convergence and reduced divergent transitions.
   • Posterior summaries represent the central tendency and uncertainty around the model parameters through credible intervals (CrI).
4. Model Formula:

diabetes_dx | weights(wt_norm) ~ age_c + bmi_c + sex + race

Family: bernoulli Links: mu = logit Data: adult_imp1 (Number of observations: 5592) Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1; total post-warmup draws = 4000 Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS and Tail_ESS are effective sample size measures, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat = 1).

library(gt)

priors <- c(
  set_prior("normal(0, 2.5)", class = "b"),
  set_prior("student_t(3, 0, 10)", class = "Intercept") 
)

bayes_fit <- brm(
  formula = diabetes_dx | weights(wt_norm) ~ age_c + bmi_c + sex + race,
  data    = adult_imp1,
  family  = bernoulli(link = "logit"),
  prior   = priors,
  chains  = 4, iter = 2000, seed = 123,
  control = list(adapt_delta = 0.95),
  refresh = 0   # quiet Stan output
)

Running /opt/R/4.4.2/lib/R/bin/R CMD SHLIB foo.c
using C compiler: ‘gcc (GCC) 11.5.0 20240719 (Red Hat 11.5.0-2)’
gcc -I"/opt/R/4.4.2/lib/R/include" -DNDEBUG   -I"/opt/R/4.4.2/lib/R/library/Rcpp/include/"  -I"/opt/R/4.4.2/lib/R/library/RcppEigen/include/"  -I"/opt/R/4.4.2/lib/R/library/RcppEigen/include/unsupported"  -I"/opt/R/4.4.2/lib/R/library/BH/include" -I"/opt/R/4.4.2/lib/R/library/StanHeaders/include/src/"  -I"/opt/R/4.4.2/lib/R/library/StanHeaders/include/"  -I"/opt/R/4.4.2/lib/R/library/RcppParallel/include/"  -I"/opt/R/4.4.2/lib/R/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DUSE_STANC3 -DSTRICT_R_HEADERS  -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION  -D_HAS_AUTO_PTR_ETC=0  -include '/opt/R/4.4.2/lib/R/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include    -fpic  -g -O2  -c foo.c -o foo.o
In file included from /opt/R/4.4.2/lib/R/library/RcppEigen/include/Eigen/Core:19,
                 from /opt/R/4.4.2/lib/R/library/RcppEigen/include/Eigen/Dense:1,
                 from /opt/R/4.4.2/lib/R/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22,
                 from <command-line>:
/opt/R/4.4.2/lib/R/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: cmath: No such file or directory
  679 | #include <cmath>
      |          ^~~~~~~
compilation terminated.
make: *** [/opt/R/4.4.2/lib/R/etc/Makeconf:195: foo.o] Error 1

prior_summary(bayes_fit)

               prior     class                coef group resp dpar nlpar lb ub
      normal(0, 2.5)         b                                                
      normal(0, 2.5)         b               age_c                            
      normal(0, 2.5)         b               bmi_c                            
      normal(0, 2.5)         b raceMexicanAmerican                            
      normal(0, 2.5)         b         raceNHBlack                            
      normal(0, 2.5)         b     raceOtherDMulti                            
      normal(0, 2.5)         b   raceOtherHispanic                            
      normal(0, 2.5)         b           sexFemale                            
 student_t(3, 0, 10) Intercept                                                
 tag       source
             user
     (vectorized)
     (vectorized)
     (vectorized)
     (vectorized)
     (vectorized)
     (vectorized)
     (vectorized)
             user

summary(bayes_fit)            # Bayesian model summary

 Family: bernoulli 
  Links: mu = logit 
Formula: diabetes_dx | weights(wt_norm) ~ age_c + bmi_c + sex + race 
   Data: adult_imp1 (Number of observations: 5592) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Regression Coefficients:
                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept              -2.66      0.09    -2.84    -2.50 1.00     4187     3510
age_c                   1.09      0.06     0.97     1.22 1.00     3012     3098
bmi_c                   0.63      0.05     0.53     0.72 1.00     3472     3315
sexFemale              -0.66      0.10    -0.86    -0.46 1.00     4003     3052
raceMexicanAmerican     0.69      0.18     0.35     1.04 1.00     3526     2843
raceOtherHispanic       0.43      0.24    -0.07     0.89 1.00     4058     3114
raceNHBlack             0.54      0.15     0.24     0.82 1.00     3597     3177
raceOtherDMulti         0.82      0.19     0.45     1.19 1.00     3763     3257

Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

Results and Visulaization

Results of Bayesian Logistic Regression: - All Rhat ≈ 1.00 → model convergence is excellent - ESS values are large (>1000), indicating stable posterior sampling and indicate model reliability. - Baseline log-odds of diabetes for a male, non-Hispanic White individual with mean BMI and age → corresponds to very low probability of diabetes (~7%). - For each 1 SD increase in age, the odds of diabetes are about 3 times higher, with high certainty (credible interval well above 0). - Each 1 SD increase in BMI is associated with an 88% increase in odds of diabetes — a strong, significant effect. - Females have 48% lower odds of diabetes than males, credible interval excludes 0 → strong evidence. - Mexican Americans have about 2× higher odds of diabetes compared with Non-Hispanic Whites. Credible interval is entirely >0 → significant. - Other Hispanics have slightly higher odds (≈1.5×), but the credible interval overlaps 0 → uncertain evidence - Non-Hispanic Blacks have ~1.7× higher odds, with credible interval >0 → significant association. - Individuals identifying as Other/Multi-racial have more than double the odds of diabetes, strong evidence.

1. Plot: Distributions for Coefficients for age and bmi

• The density plots of standardized BMI and Age from the imputed dataset (adult_imp1) show approximately normal distributions centered near zero, consistent with z-score standardization confirming that both predictors were properly centered and scaled prior to Bayesian modeling, ensuring comparability and numerical stability during estimation.

2. Plot: Prior Distributions for Coefficients

• Priors for regression coefficients were drawn from a Normal(0, 2.5) distribution, representing weakly informative assumptions centered at zero with moderate spread. The prior density plots for Age (per 1 SD) and BMI (per 1 SD) demonstrate symmetric bell-shaped distributions, indicating no strong bias toward positive or negative effects before observing data.

library(ggplot2)

# adult_imp1 plot 

# Convert to long format
adult_long <- adult_imp1 %>%
  select(bmi_c, age_c) %>%
  pivot_longer(cols = everything(), names_to = "Coefficient", values_to = "Value")

# Plot
ggplot(adult_long, aes(x = Value, fill = Coefficient)) +
  geom_density(alpha = 0.5) +
  theme_minimal() +
  labs(title = "Distributions for Coefficients from adult_imp1 data",
       x = "Coefficient Value", y = "Density") +
  scale_fill_manual(values = c("bmi_c" = "skyblue", "age_c" = "orange"))

## prior draws 

prior_draws <- tibble(
  term = rep(c("Age (per 1 SD)", "BMI (per 1 SD)"), each = 4000),
  value = c(rnorm(4000, 0, 2.5), rnorm(4000, 0, 2.5))
)

## Plot (prior) (age and bmi) 
ggplot(prior_draws, aes(x = value, fill = term)) +
  geom_density(alpha = 0.5) +
  theme_minimal() +
  labs(title = "Prior Distributions for Coefficients",
       x = "Coefficient Value", y = "Density") +
  scale_fill_manual(values = c("skyblue", "orange"))

library(brms)

plot(bayes_fit)   # Posterior distributions

Predictive Checking

MCMC Trace Plots (Right Panels)

• Each shows 4 MCMC chains (different colors) across 1000 iterations:
• The chains mix well and overlap substantially, without visible trends or drifts → indicates good convergence.
• The parameter values oscillate around stable means with no systematic pattern → confirms stationarity.
• Combined with Rhat ≈ 1 and high ESS from your summary, the trace plots visually validate posterior convergence and independence.

MCMC Convergence: MCMC Trace plots show chains for each parameter over iterations are well-mixed chains without trends indicate convergence and stable posterior estimates.

Posterior Distributions (Left Panels)

• plots visualizes posterior distributions with high uncertainty, narrow distributions indicating precise estimates.

• All distributions look smooth and unimodal → no multimodality, confirming stable posteriors.

• Each histogram represents the distribution of sampled coefficient values after convergence across all MCMC draws:

• b_raceOtherHispanic: The posterior peaks around 0.4–0.5, with some spread below 0 and above 1.→ Suggests a modestly positive association with diabetes risk, but some uncertainty (credible interval overlaps 0).

• b_raceNHBlack: Centered around 0.5–0.6, with a narrower, symmetric shape. → Indicates a consistent positive effect—NH Black participants have higher odds of diabetes, and uncertainty is low.

• b_raceOtherDMulti: Centered around 0.8–0.9, with slightly wider spread but entirely above 0. → Stronger evidence for increased odds of diabetes among Other/Multi-racial individuals.

Results from Posterior

• The Bayesian logistic regression model identified significant associations between demographic and anthropometric factors and diabetes diagnosis.
• Below is the tabulated format for Bayesian posterior odds ratios (95% CrI) — reference: NH White (race), Male (sex)
• Age a strong predictor: each standardized unit increase in age was associated with nearly threefold higher odds of diabetes (OR = 2.99; 95% CrI = 2.64–3.37).
• BMI showed a strong positive association (OR = 1.87; 95% CrI = 1.71–2.05), higher body mass substantially increased diabetes risk.
• Female sex had lower odds of diabetes compared to males (OR = 0.52; 95% CrI = 0.42–0.63).
• Compared with Non-Hispanic Whites (reference group), several racial/ethnic groups had higher odds:
• Mexican Americans (OR = 2.00; 95% CrI = 1.41–2.84)
• Non-Hispanic Blacks (OR = 1.71; 95% CrI = 1.28–2.27)
• Other/Multi-racial individuals (OR = 2.27; 95% CrI = 1.56–3.28)
• Other Hispanics showed a positive but non-significant association (OR = 1.54; 95% CrI = 0.93–2.43).

# Posterior ORs (drop intercept, clean labels)

bayes_or <- posterior_summary(bayes_fit, pars = "^b_") %>%
  as.data.frame() %>%
  tibble::rownames_to_column("raw") %>%
  dplyr::mutate(
    term = gsub("^b_", "", raw),
    term = gsub("race", "race:", term),
    term = gsub("sex",  "sex:",  term),
    term = gsub("OtherDMulti", "Other/Multi", term),
    term = gsub("OtherHispanic", "Other Hispanic", term),
    OR   = exp(Estimate),
    LCL  = exp(Q2.5),
    UCL  = exp(Q97.5)
  ) %>%
  dplyr::select(term, OR, LCL, UCL) %>%
  dplyr::filter(term != "Intercept")

knitr::kable(
  bayes_or %>%
    dplyr::mutate(dplyr::across(c(OR,LCL,UCL), ~round(.x, 2))),
  digits = 2,
  caption = "Bayesian posterior odds ratios (95% CrI) — reference: NH White (race), Male (sex)"
)

Bayesian posterior odds ratios (95% CrI) — reference: NH White (race), Male (sex)

term	OR	LCL	UCL
age_c	2.99	2.64	3.37
bmi_c	1.87	1.71	2.05
sex:Female	0.52	0.42	0.63
race:MexicanAmerican	2.00	1.41	2.84
race:Other Hispanic	1.54	0.93	2.43
race:NHBlack	1.71	1.28	2.27
race:Other/Multi	2.27	1.56	3.28

library(brms)
library(dplyr)
library(posterior)
library(bayesplot)
library(ggplot2)


# Extract posterior draws as a draws_df # simulate posterior outcomes
post <- as_draws_df(bayes_fit)
post

# A draws_df: 1000 iterations, 4 chains, and 11 variables
   b_Intercept b_age_c b_bmi_c b_sexFemale b_raceMexicanAmerican
1         -2.6     1.1    0.70       -0.71                  0.67
2         -2.7     1.0    0.62       -0.57                  0.65
3         -2.6     1.1    0.65       -0.76                  0.63
4         -2.7     1.0    0.65       -0.67                  0.82
5         -2.6     1.1    0.61       -0.73                  0.75
6         -2.5     1.0    0.60       -0.77                  0.61
7         -2.8     1.1    0.66       -0.66                  0.52
8         -2.8     1.2    0.67       -0.57                  0.94
9         -2.8     1.1    0.65       -0.52                  0.84
10        -2.6     1.1    0.67       -0.85                  0.70
   b_raceOtherHispanic b_raceNHBlack b_raceOtherDMulti
1                0.605          0.52              0.95
2                0.338          0.45              0.69
3                0.566          0.63              0.54
4                0.453          0.61              0.78
5                0.090          0.50              0.62
6                0.015          0.48              0.60
7                0.736          0.50              0.84
8                0.913          0.57              1.07
9                0.570          0.66              0.81
10               0.467          0.54              0.97
# ... with 3990 more draws, and 3 more variables
# ... hidden reserved variables {'.chain', '.iteration', '.draw'}

# Posterior summary
post_sum <- posterior_summary(bayes_fit)
post_sum

                           Estimate  Est.Error          Q2.5         Q97.5
b_Intercept              -2.6644912 0.08841987 -2.840368e+00    -2.4963936
b_age_c                   1.0936287 0.06218042  9.720147e-01     1.2162894
b_bmi_c                   0.6267208 0.04755308  5.344481e-01     0.7198901
b_sexFemale              -0.6586208 0.10156670 -8.592825e-01    -0.4575489
b_raceMexicanAmerican     0.6916992 0.17744231  3.465940e-01     1.0425831
b_raceOtherHispanic       0.4314438 0.24442275 -7.159507e-02     0.8867589
b_raceNHBlack             0.5379213 0.14668730  2.431286e-01     0.8182113
b_raceOtherDMulti         0.8190024 0.18868276  4.454199e-01     1.1877554
Intercept                -2.6732989 0.06779455 -2.808774e+00    -2.5477114
lprior                  -16.5021561 0.05307875 -1.661688e+01   -16.4105283
lp__                  -1430.3473284 2.03852832 -1.435302e+03 -1427.4169785

Posterior Summary

• Posterior Summaries included: mean, median, 95% credible intervals
• Convergence diagnostics (R-hat, effective sample size).
  -Posterior Odds Ratios interpreting model coefficients with reference categories: NH White (race), Male (sex).
• Estimate (posterior mean of the coefficient) representing the central tendency of the parameter’s distribution.
• Est.Error (posterior standard deviation) quantifying uncertainty around the estimate.
• The 2.5th percentile of the posterior distribution, representing the lower bound of the 95% credible interval.
• The 97.5th percentile of the posterior distribution, representing the upper bound of the 95% credible interval.

Posterior Distributions of Model Coefficients

• Using 4,000 posterior draws from the Bayesian model (4 chains × 1,000 post-warmup draws per chain), the mcmc_areas() plot visualized the posterior distributions of key predictors: age, BMI, sex, and race/ethnicity.
• The posterior densities show the range and uncertainty of each coefficient.
• The 95% credible intervals are clearly depicted by the shaded areas, highlighting which predictors have strong evidence of association with diabetes.
• The Bayesian model produced posterior estimates for the effects of BMI and age (standardized per 1 SD) on the outcome. Age and BMI showed positive associations
  • BMI showed a negative association in most draws, with posterior estimates ranging roughly from 0.61 to 0.70, indicating that higher BMI is associated with lower odds of the outcome in this analysis.
  • Age showed a positive association, with posterior estimates ranging roughly from 1.00 to 1.14, suggesting that higher age increases the odds of the outcome.
  • female sex showed a negative association
  • Racial/ethnic groups had elevated odds relative to the reference group.

# Posterior predictive draws

#Posterior predictive checks (binary outcome)
pp_samples <- posterior_predict(bayes_fit, ndraws = 500)  # 500 draws

# Check dimensions
dim(pp_samples)  # rows = draws, cols = observations

[1]  500 5592

Posterior Predictive Checks (PPC)

• To assesses how the model reproduces observed data and validate model fit, visualizations below compares the predicted and observed data to show density overlays for both mean and SD. There was no large discrepancies indicating potential misfit and good alignment suggesting reliable predictions.
• A total of 50 posterior predictive sample draws generated 5,592 observations, producing a simulated distribution of predicted outcomes consistent with the sample size and assesses model fit and evaluated how well the Bayesian model reproduced the observed data pattern.
• ppc_bars() plot compared the observed diabetes outcomes with 50 replicated datasets drawn from the posterior distribution.
• The replicated distributions closely matched the observed proportions, indicating that the Bayesian model adequately captured the outcome variability and overall data structure.
  
• Model’s predictions align with reality where mean(y_rep) = average predicted probability of diabetes for each individual, across all posterior draws of the parameters and y = the actual observed diabetes status (0 = non-diabetic, 1 = diabetic).

ppc_bars(y = adult_imp1$diabetes_dx, yrep = pp_samples[1:50, ])

#PP check for proportions (useful for binary) mean comparison to check if the simulated means match the observed mean

## mean
ppc_stat(y = adult_imp1$diabetes_dx, yrep = pp_samples[1:100, ], stat = "mean") +
  labs(title = "Posterior Predictive Check: Mean of Replicates") +
  theme_minimal()

Posterior Predictive stats: Mean

-A posterior predictive check was performed on the mean diabetes outcome using 100 replicated datasets from the posterior distribution. - The distribution of the simulated means closely aligned with the observed mean, suggesting that the Bayesian model accurately captures the central tendency of the outcome.

#PP check for proportions (useful for binary) mean and sd comparison to check if the simulated means match the observed mean

## sd
ppc_stat(y = adult_imp1$diabetes_dx, yrep = pp_samples[1:100, ], stat = "sd") +
  labs(title = "PPC: Standard Deviation of Replicates") +
  theme_minimal()

Posterior Predictive stats: Standard Deviation

• A posterior predictive check was conducted on the standard deviation of diabetes outcomes using 100 replicated datasets from the posterior distribution.
• The simulated standard deviations closely matched the observed value, indicating that the Bayesian model adequately captures the variability in the outcome data.

library(brms)
library(dplyr)
library(posterior)
library(bayesplot)
library(ggplot2)


post_wide <- as_draws_df(bayes_fit)  # or posterior_samples(bayes_fit)
names(post_wide)  # should show: b_Intercept, b_bmi_c, b_age_c, etc.

 [1] "b_Intercept"           "b_age_c"               "b_bmi_c"              
 [4] "b_sexFemale"           "b_raceMexicanAmerican" "b_raceOtherHispanic"  
 [7] "b_raceNHBlack"         "b_raceOtherDMulti"     "Intercept"            
[10] "lprior"                "lp__"                  ".chain"               
[13] ".iteration"            ".draw"                

mcmc_areas(
  post_wide,
  pars = c("b_age_c","b_bmi_c","b_sexFemale",
           "b_raceMexicanAmerican","b_raceOtherHispanic",
           "b_raceNHBlack","b_raceOtherDMulti")
)

Posterior Distributions of Regression Coefficients for Predictors of Diabetes

This figure presents the posterior distributions of regression coefficients estimated from the Bayesian logistic regression model. Each density curve represents the uncertainty around the parameter’s posterior mean for a given predictor.

• Age and BMI show strong positive associations with diabetes status, as their posterior distributions are concentrated above zero, suggesting higher values increase the likelihood of diabetes.

• • Sex (Female) has a distribution centered slightly below zero, indicating a lower probability of diabetes compared to males.

  • Race categories (Mexican American, Other Hispanic, Non-Hispanic Black, and Other/Multi) show broader distributions with varying levels of uncertainty relative to the reference group (Non-Hispanic White).

  • The shaded regions indicate the 80% credible intervals, representing the range within which the true parameter values are most likely to lie based on the posterior samples.

bayes_R2(bayes_fit)      # Model fit

    Estimate  Est.Error      Q2.5    Q97.5
R2 0.1313342 0.01265055 0.1064607 0.156078

Bayesian 𝑅^2 (model fit statics):

Model Fit to quantify predictive performance. - Explains about 13% of the variability in diabetes status, with credible uncertainty bounds suggesting reasonable but modest explanatory power. - but other factors (e.g., genetics, lifestyle, environment) also contribute to outcome variability.

predicted <- fitted(bayes_fit, summary = TRUE)
observed <- adult_imp1[, c("bmi", "age")]

# Plot for **bmi** (obs vs pred)

ggplot(data = NULL, aes(x = observed$bmi, y = predicted[, "Estimate"])) +
  geom_point() +
  geom_errorbar(aes(ymin = predicted[, "Q2.5"], ymax = predicted[, "Q97.5"])) +
  geom_abline(slope = 1, intercept = 0, linetype = "dashed", color = "red") +
  xlab("Observed bmi") + ylab("Predicted bmi")

Observed vs. Predicted BMI

• A comparison of observed BMI values with their posterior predicted estimates was performed using the Bayesian model.
• Each point represents an individual’s observed BMI versus the model’s predicted mean.
• Error bars indicate the 95% credible intervals of the predictions.
• The dashed red line represents perfect prediction (observed = predicted). - The plot demonstrates that the model’s predictions generally align with the observed data, with most points closely following the diagonal, indicating good predictive performance for BMI.

library(posterior)
library(dplyr)
library(tidyr)

# Extract posterior draws as a matrix, then convert to tibble
post <- as_draws_matrix(bayes_fit) %>%   # safer than as_draws_df for manipulation
  as.data.frame() %>%
  select(b_bmi_c, b_age_c) %>%
  pivot_longer(
    everything(),
    names_to = "term",
    values_to = "estimate"
  ) %>%
  mutate(
    term = case_when(
      term == "b_bmi_c" ~ "BMI (per 1 SD)",
      term == "b_age_c" ~ "Age (per 1 SD)"
    ),
    type = "Posterior"
  )
prior_draws <- tibble(
  term = rep(c("BMI (per 1 SD)", "Age (per 1 SD)"), each = 4000),
  estimate = c(rnorm(4000, 0, 1), rnorm(4000, 0, 1)),
  type = "Prior"
)
combined_draws <- bind_rows(prior_draws, post)

library(ggplot2)

ggplot(combined_draws, aes(x = estimate, fill = type)) +
  geom_density(alpha = 0.4) +
  facet_wrap(~ term, scales = "free", ncol = 2) +
  theme_minimal(base_size = 13) +
  labs(
    title = "Prior vs Posterior Distributions",
    x = "Coefficient estimate",
    y = "Density",
    fill = ""
  )

Prior vs Posterior Distributions Visualization:

Density plots were generated to overlay the prior and posterior distributions for BMI and age. Posterior distributions were narrower and shifted away from zero relative to the priors, indicating that the data provided substantial evidence for positive associations of BMI and age with diabetes risk. Faceting by predictor enabled clear visual comparison of how each coefficient’s distribution evolved after model updating.

Interpretation: The shift from diffuse priors to concentrated posteriors demonstrates the model’s ability to incorporate empirical evidence and refine prior beliefs. Narrower posterior distributions reflect reduced uncertainty in parameter estimates. The posterior predictive proportions of diabetes closely matched the observed prevalence in the analytic dataset, supporting the model’s reliability for inference and prediction.

Model Comparison

• Model Comparison Across all analytic methods: survey-weighted maximum likelihood estimation (MLE), multiple imputation with pooled estimates (MICE), and Bayesian regression shows the associations between BMI, age, and diabetes diagnosis were consistent in direction and magnitude.

• For BMI, the odds ratios ranged from 1.73 (95% CI: 1.58–1.89) in the MICE-pooled model to 1.89 (95% CI: 1.65–2.15) in the survey-weighted MLE model, and 1.87 (95% CrI: 1.71–2.05) in the Bayesian model.

• For age, the estimated odds ratios were 2.90 (95% CI: 2.60–3.24) using MICE, 3.03 (95% CI: 2.70–3.40) from the survey-weighted MLE model, and 2.99 (95% CrI: 2.64–3.37) in the Bayesian analysis.

# Results

 #Build compact results table (BMI & Age only) 
library(dplyr); 
library(tidyr); 
library(knitr); 
library(stringr)

# pretty "OR (LCL–UCL)" string

  fmt_or <- function(or, lcl, ucl, digits = 2) {
  paste0(
    formatC(or,  format = "f", digits = digits), " (",
    formatC(lcl, format = "f", digits = digits), "–",
    formatC(ucl, format = "f", digits = digits), ")"
  )
}

# guardrails: require these to exist from Modeling
stopifnot(exists("svy_or"), exists("mi_or"), exists("bayes_or"))
for (nm in c("svy_or","mi_or","bayes_or")) {
  if (!all(c("term","OR","LCL","UCL") %in% names(get(nm)))) {
    stop(nm, " must have columns: term, OR, LCL, UCL")
  }
}

svy_tbl   <- svy_or   %>% mutate(Model = "Survey-weighted MLE")
mi_tbl    <- mi_or    %>% mutate(Model = "mice pooled")
bayes_tbl <- bayes_or %>% mutate(Model = "Bayesian")

all_tbl <- bind_rows(svy_tbl, mi_tbl, bayes_tbl) %>%
  mutate(term = case_when(
    str_detect(term, "bmi_c|\\bBMI\\b") ~ "BMI (per 1 SD)",
    str_detect(term, "age_c|\\bAge\\b") ~ "Age (per 1 SD)",
    TRUE ~ term
  )) %>%
  filter(term %in% c("BMI (per 1 SD)", "Age (per 1 SD)")) %>%
  mutate(OR_CI = fmt_or(OR, LCL, UCL, digits = 2)) %>%
  select(Model, term, OR_CI) %>%
  arrange(
    factor(Model, levels = c("Survey-weighted MLE","mice pooled","Bayesian")),
    factor(term,  levels = c("BMI (per 1 SD)","Age (per 1 SD)"))
  )

res_wide <- all_tbl %>%
  pivot_wider(names_from = term, values_from = OR_CI) %>%
  rename(
    `BMI (per 1 SD) OR (95% CI)` = `BMI (per 1 SD)`,
    `Age (per 1 SD) OR (95% CI)` = `Age (per 1 SD)`
  )

kable(
  res_wide,
  align = c("l","c","c"),
  caption = "Odds ratios (per 1 SD) with 95% CIs across models"
)

Odds ratios (per 1 SD) with 95% CIs across models

Model	BMI (per 1 SD) OR (95% CI)	Age (per 1 SD) OR (95% CI)
Survey-weighted MLE	1.89 (1.65–2.15)	3.03 (2.70–3.40)
mice pooled	1.73 (1.58–1.89)	2.90 (2.60–3.24)
Bayesian	1.87 (1.71–2.05)	2.99 (2.64–3.37)

# Compute proportion of diabetes=1 for each draw
pp_proportion <- rowMeans(pp_samples)  # proportion of 1's in each posterior draw

# Optional: visualize the posterior probability distribution
pp_proportion_df <- tibble(proportion = pp_proportion)

ggplot(pp_proportion_df, aes(x = proportion)) +
  geom_histogram(binwidth = 0.01, fill = "skyblue", color = "black") +
  labs(
    title = "Posterior Distribution of Proportion of Diabetes = 1",
    x = "Proportion of Diabetes = 1",
    y = "Frequency"
  ) +
  theme_minimal()

Posterior Distribution of Diabetes Prevalence

• A histogram of these values shows the distribution of predicted prevalence calculated for each draw of the Bayesian model, reflecting uncertainty in the model estimates, central tendency and variability of diabetes prevalence
• Most posterior predictions cluster around 10–11%, indicating good alignment with the observed/imputed data and demonstrating that the model captures the underlying population pattern.

library(dplyr)
library(knitr)
svy_mean <- svymean(~diabetes_dx, nhanes_design_adult, na.rm = TRUE)

# Create summary table
summary_table <- tibble(
  Method = c("Survey-weighted mean (NHANES)", 
             "Imputed dataset mean", 
             "Posterior predictive mean"),
  diabetes_mean = c(
    coef(svy_mean),           # survey-weighted mean
    mean(adult_imp1$diabetes_dx, na.rm = TRUE),  # imputed dataset
    mean(pp_proportion)       # posterior predictive mean
  ),
  SE = c(
    SE(svy_mean),             # survey-weighted SE
    NA,                       # not available for raw mean
    NA                        # not available for posterior predictive
  )
)

# Render table
kable(summary_table, digits = 4, caption = "Comparison of Diabetes Prevalence Across Methods")

Comparison of Diabetes Prevalence Across Methods

Method	diabetes_mean	SE
Survey-weighted mean (NHANES)	0.0890	0.0048
Imputed dataset mean	0.1105	NA
Posterior predictive mean	0.1089	NA

Comparison of Diabetes Prevalence Across Methods

• The estimated prevalence of diabetes was consistent across different analytical approaches:

• Survey-weighted mean (NHANES): 8.9% (SE = 0.0048)

• Imputed dataset mean (MICE): 11.1%

• Posterior predictive mean (Bayesian model): 10.95%

  These results indicate that multiple imputation and Bayesian posterior predictions yield slightly higher prevalence estimates than the raw survey-weighted mean, but all methods are broadly consistent. The posterior predictive distribution closely matches the observed prevalence, suggesting that the Bayesian model is well-calibrated.

• Bayesian model predicts that about 10–11% of this population has diabetes, with a relatively narrow range across posterior draws, reflects uncertainty in the estimate

• While most predictions cluster around 10–11%, the model allows for values as low as 8.5% and as high as 12.8%.

• On comparing this with the raw imputed data proportion show that the the model predictions align with the observed/imputed data.

  The predicted proportion incorporates uncertainty from both the Bayesian model and the imputed data, providing a more robust estimate of diabetes prevalence.

  These results suggest that approximately 1 in 10 adults in this population may have diabetes, which can help policymakers and clinicians plan and prioritize targeted interventions effectively.

library(tidyverse)

# Posterior predicted proportion vector
# pp_proportion <- rowMeans(pp_samples)  # if not already done

known_prev <- 0.089   # NHANES prevalence

# Posterior summary
posterior_mean <- mean(pp_proportion)
posterior_ci <- quantile(pp_proportion, c(0.025, 0.975))  # 95% credible interval

# Create a data frame for plotting
pp_df <- tibble(proportion = pp_proportion)

# Plot
ggplot(pp_df, aes(x = proportion)) +
  geom_histogram(binwidth = 0.005, fill = "skyblue", color = "black") +
  geom_vline(xintercept = known_prev, color = "red", linetype = "dashed", size = 1) +
  geom_vline(xintercept = posterior_mean, color = "blue", linetype = "solid", size = 1) +
  geom_rect(aes(xmin = posterior_ci[1], xmax = posterior_ci[2], ymin = 0, ymax = Inf),
            fill = "blue", alpha = 0.1, inherit.aes = FALSE) +
  labs(
    title = "Posterior Predicted Diabetes Proportion vs NHANES Prevalence",
    subtitle = paste0("Red dashed = NHANES prevalence (", known_prev, 
                      "), Blue solid = Posterior mean (", round(posterior_mean,3), ")"),
    x = "Proportion of Diabetes = 1",
    y = "Frequency"
  ) +
  theme_minimal()

Posterior Predicted Diabetes Proportion vs. NHANES Prevalence

• We compared the posterior predicted proportion of diabetes from the Bayesian model with the observed NHANES prevalence (8.9%).
• The blue solid line represents the posterior mean, while the shaded blue area indicates the 95% credible interval of predicted proportions.
• The red dashed line shows the NHANES survey-weighted prevalence for reference.
• Most posterior predictions cluster around 10–11%, slightly higher than the NHANES mean, but the credible interval overlaps the observed prevalence, indicating good model calibration.
• This visualization highlights the model’s ability to capture uncertainty in predictions while remaining consistent with the observed data.

library(dplyr)

# Posterior predicted proportion

posterior_mean <- mean(pp_proportion)
posterior_ci <- quantile(pp_proportion, c(0.025, 0.975))  # 95% credible interval

# NHANES prevalence with SE from survey::svymean
# Suppose you already have:
# svymean(~diabetes_dx, nhanes_design_adult, na.rm = TRUE)
known_prev <- 0.089        # Mean prevalence
known_se   <- 0.0048       # Standard error from survey

# Calculate 95% confidence interval
known_ci <- c(
  known_prev - 1.96 * known_se,
  known_prev + 1.96 * known_se
)

# Print results
data.frame(
  Type = c("Posterior Prediction", "NHANES Prevalence"),
  Mean = c(posterior_mean, known_prev),
  Lower_95 = c(posterior_ci[1], known_ci[1]),
  Upper_95 = c(posterior_ci[2], known_ci[2])
)

                     Type      Mean   Lower_95  Upper_95
2.5% Posterior Prediction 0.1089181 0.09629381 0.1216962
        NHANES Prevalence 0.0890000 0.07959200 0.0984080

# Create a data frame for plotting
ci_df <- data.frame(
  Type = c("Posterior Prediction", "NHANES Prevalence"),
  Mean = c(0.1096674, 0.089),
  Lower_95 = c(0.09772443, 0.079592),
  Upper_95 = c(0.1210658, 0.098408)
)


# --- 1. Survey-weighted (Population) prevalence ---
pop_est <- svymean(~diabetes_dx, nhanes_design_adult, na.rm = TRUE)
pop_prev <- as.numeric(pop_est)
pop_se <- as.numeric(SE(pop_est))
pop_ci <- c(pop_prev - 1.96 * pop_se, pop_prev + 1.96 * pop_se)

# --- 2. Bayesian posterior prevalence ---
# bayes_pred = matrix of posterior draws (iterations × individuals)
pp_proportion <- rowMeans(pp_samples)             # prevalence per posterior draw
post_prev <- mean(pp_proportion)                  # posterior mean prevalence
post_ci <- quantile(pp_proportion, c(0.025, 0.975))  # 95% credible interval

# --- 3. Combine into one data frame ---
bar_df <- tibble(
  Source     = c("Survey-weighted (Population)", "Bayesian Posterior"),
  Prevalence = c(pop_prev, post_prev),
  CI_low     = c(pop_ci[1], post_ci[1]),
  CI_high    = c(pop_ci[2], post_ci[2])
)

# --- 4. Plot ---
ggplot(bar_df, aes(x = Source, y = Prevalence, fill = Source)) +
  geom_col(alpha = 0.85, width = 0.6) +
  geom_errorbar(
    aes(ymin = CI_low, ymax = CI_high),
    width = 0.15,
    color = "black",
    linewidth = 0.8
  ) +
  guides(fill = "none") +
  labs(
    title = "Population vs Posterior Diabetes Prevalence",
    subtitle = "Survey-weighted estimate (design-based) vs Bayesian (model-based)",
    y = "Prevalence (Proportion with Diabetes)",
    x = NULL
  ) +
  theme_minimal(base_size = 13)

Comparison of Diabetes Prevalence Across Methods

• The bar plot compares survey-weighted, imputed, and Bayesian posterior estimates of diabetes prevalence:
• Survey-weighted prevalence: 8.9% (95% CI: 0.080–0.098), reflecting the NHANES population after accounting for complex sampling.
• Imputed (unweighted) prevalence: 11.1%, slightly higher due to unadjusted overrepresentation of subgroups with higher diabetes rates.
• Bayesian posterior mean: 10.9% (95% CrI: 0.098–0.121), closely replicating the imputed data while slightly shrinking toward the population-level mean, consistent with Bayesian regularization.
• The posterior credible interval overlaps the survey 95% CI, indicating that the Bayesian model reproduces population-level prevalence accurately. This demonstrates good model calibration and predictive validity, while visualizing the uncertainty of both survey-based and model-based estimates.

Practical Implications - Health departments can estimate diabetes burden at the state or county level using Bayesian small-area estimation. - Clinicians and public health researchers can plan targeted screening where predicted prevalence is higher than observed. - Epidemiologists can validate disease models before applying them to regions without survey data.

library(tidyr)
library(bayesplot)
library(posterior)

# Convert fitted model to draws array
post_array <- as_draws_array(bayes_fit)  # draws x chains x parameters

# Plot autocorrelation for age and bmi
mcmc_acf(post_array, pars = c("b_age_c", "b_bmi_c"))

MCMC Autocorrelation for Key Parameters

Pairwise plots (mcmc_pairs, posterior) explore correlations between parameters. - Autocorrelation plots for the posterior draws of age and BMI coefficients assess chain mixing and convergence showing correlation of each draw with its lagged values across iterations. - Rapid decay of autocorrelation toward zero indicates that the Markov chains are mixing well and successive draws are relatively independent. - Both age and BMI coefficients exhibited low autocorrelation after a few lags, supporting the reliability of posterior estimates. - This diagnostic confirms that the Bayesian model sampling was adequate and stable, ensuring valid inference from the posterior distributions.

Conclusion

Model Performance and Fit

• Across multiple modeling approaches — survey-weighted MLE, multiple imputation (MICE), and Bayesian logistic regression — the models were consistent and reliable in predicting diabetes risk.
• The Bayesian model showed excellent convergence (Rhat = 1.00; Bulk/Tail ESS > 2000) and explained approximately 13% of the variance in diabetes (Posterior R² = 0.13, 95% CrI: 0.106–0.156).
• Posterior predictive checks confirmed good model calibration, with credible intervals capturing predictive uncertainty.

Key Predictors

• Age and BMI were the strongest and most consistent predictors across all methods. Each 1 SD increase in age nearly tripled the odds of diabetes, while a 1 SD increase in BMI increased the odds by approximately 1.7–1.9×.
• Sex (female) was associated with lower odds of diabetes.
• Race/Ethnicity: Mexican American, NH Black, and Other/Multi groups had significantly higher odds; the effect for Other Hispanic was less certain.

Interpretation and Robustness

• Bayesian credible intervals were slightly narrower than frequentist confidence intervals but largely overlapped, indicating robust effect estimates and stable inference.
• Prior regularization in the Bayesian model stabilized estimates in the presence of modest missingness.
• The results are slightly conservative relative to the observed population prevalence, but the posterior predictions remain consistent with both survey-weighted and imputed data.

Design Considerations

• Survey-weighted MLE accounts for the complex NHANES sampling design, producing population-representative estimates.
• The Bayesian model used normalized weights as importance weights, which approximates the effect of survey weights but does not fully account for stratification, clustering, or design-based variance adjustments.

Overall: - Age and BMI are consistently strong risk factors for diabetes in this population. - The Bayesian model complements frequentist approaches by providing stable, interpretable, and uncertainty-quantified estimates, while broadly reproducing population-level prevalence.

Discussions

• The use of multiple imputation allowed for robust analysis despite missing data, increasing power and reducing bias.
• Comparison of frequentist and Bayesian models demonstrated consistency in significant predictors, while Bayesian approaches provided the advantage of posterior distributions and probabilistic interpretation.
• Across all models, both age and BMI emerged as strong and consistent predictors of diabetes.
• The consistency across modeling approaches strengthens the validity of these findings Multiple imputation accounted for potential biases due to missing data, and Bayesian modeling provided robust credible intervals that closely matched frequentist estimates align with previous epidemiological research indicating that increasing age and higher BMI are among the most important determinants of type 2 diabetes risk.
• Cumulative exposure to metabolic and lifestyle risk factors over time, and the role of excess adiposity and insulin related effects account for diabetes.
• Survey weighted dataset strenghthens ensuring population representativeness, multiple imputation to handle missing data, and rigorous Bayesian estimation provided high effective sample sizes and R̂ ≈ 1.00 across parameters confirmed excellent model convergence.
• Bayesian logistic regression provided inference statistically consistent and interpretable achieving the aim of this study. In future research hierarchical model using NHANES cycles or adding variables (lab tests) could assess nonlinear effects of metabolic risk factors.

Limitations

1. The study is based on cross-sectional/observational NHANES data, which limits the ability to make causal inferences. Associations observed between BMI, age, diabetes status cannot confirm causation.
2. The project relies on multiple imputation for missing values, even though imputation reduces bias, it assumes missingness is at random (MAR); if data are missing not at random (MNAR), results may be biased.
3. Potential Residual Confounding - Models included key predictors (age, BMI, sex, race), but unmeasured factors like diet, physical activity, socioeconomic status, or genetic predisposition could confound associations.
4. Bayesian models depend on prior choices, which could influence posterior estimates if priors are informative or mis-specified.
5. Variable Measurement - BMI is measured at a single time point, which may not reflect long-term exposure or risk.
6. Self-reported variables - are subjective to recall or reporting bias.
7. Interactions are not tested in the study (bmi × age) and so other potential synergistic effects might be missed.
8. Predicted probabilities are model-based estimates, not validated for clinical decision-making. External validation in independent cohorts is needed before application.

Targeted Therapy

• Translational Perspective from the Bayesian Diabetes Prediction Project. This project further demonstrates the translational potential of Bayesian modeling in clinical decision-making and public health strategy.
• By using patient-level predictors such as age, BMI, sex, and race to estimate the probability of diabetes, the model moves beyond descriptive statistics toward individualized risk prediction.
• The translational move lies in converting these probabilistic outputs into actionable thresholds—such as identifying the BMI or age at which the predicted risk of diabetes exceeds a clinically meaningful level (e.g., 30%).
• Such insights can guide early screening, personalized lifestyle interventions, and targeted prevention programs for populations at higher risk.
• This approach embodies precision public health—bridging data science and medical decision-making to deliver tailored, evidence-based strategies that can ultimately improve diabetes prevention and management outcomes.

What changes in modifiable predictors would lower diabetes risk?

Translational Research Implications:

• We can use the model to guide prevention or intervention.
• Only BMI is a modifiable risk factor
• We can make changes in BMI (behavior or lifestyle) to achieve a lower risk threshold
• we hold non modifiable predictors as constant (sex, race).
• Vary modifiable predictors (BMI) until the model predicts the desired probability.

Internal validation

• To illustrate personalized risk estimation using the Bayesian model, we computed the posterior predicted probability of diabetes for a representative participant.
• We selected one participant from the dataset (adult[1, ]) including all relevant covariates (age, BMI, sex, race).
• Used posterior_linpred with transform = TRUE to obtain predicted probabilities for logistic regression.
• Extracted posterior draws computed 95% credible interval from the posterior draws.
• Density plot shows the distribution of plausible probabilities given the participant’s covariates.
• The density highlights uncertainty around the individual’s predicted diabetes risk.
• 95% credible interval provides a range of probable outcomes, not just a point estimate.
• This approach allows personalized risk assessment, enabling clinicians or public health practitioners to identify high-risk individuals
• Tailor preventive interventions (e.g., lifestyle modification, monitoring)
• Quantify uncertainty in predictions for decision-making
• Posterior predictive distributions enable probabilistic, individualized predictions, supporting targeted intervention strategies beyond population-level summaries.

# Use the first participant 
# using multiple covariates to select someone
participant1_data  <- adult[1, ]


# predicted probabilities for patient 1
phat1 <- posterior_linpred(bayes_fit, newdata = participant1_data, transform = TRUE)
# 'transform = TRUE' gives probabilities for logistic regression

# Store in a data frame for plotting
post_pred_df <- data.frame(pred = phat1)

# Compute 95% credible interval
ci_95_participant1 <- quantile(phat1, c(0.025, 0.975))

# Plot

ggplot(post_pred_df, aes(x = pred)) + 
  geom_density(color='darkblue', fill='lightblue') +
  geom_vline(xintercept = ci_95_participant1[1], color='red', linetype='dashed') +
  geom_vline(xintercept = ci_95_participant1[2], color='red', linetype='dashed') +
  xlab('Probability of being diabetic (Outcome=1)') +
  ggtitle('Posterior Predictive Distribution 95% Credible Interval') +
  theme_bw()

participant2_data  <- adult[2, ]


# predicted probabilities for patient 1
phat2 <- posterior_linpred(bayes_fit, newdata = participant2_data, transform = TRUE)
# 'transform = TRUE' gives probabilities for logistic regression

# Store in a data frame for plotting
post_pred_df2 <- data.frame(pred = phat2)

# Compute 95% credible interval
ci_95_participant2 <- quantile(phat2, c(0.025, 0.975))

# Plot

ggplot(post_pred_df2, aes(x = pred)) + 
  geom_density(color='darkblue', fill='lightblue') +
  geom_vline(xintercept = ci_95_participant2[1], color='red', linetype='dashed') +
  geom_vline(xintercept = ci_95_participant2[2], color='red', linetype='dashed') +
  xlab('Probability of being diabetic (Outcome=1)') +
  ggtitle('Posterior Predictive Distribution 95% Credible Interval') +
  theme_bw()

External validation

• Predicting Diabetes Risk for a New Participant to demonstrate the application of the Bayesian model for personalized prediction, we applied the trained model to a new participant not included in the original dataset.
• Selected a new participant with specific covariates (age, BMI, sex, race).
• Used posterior_linpred with transform = TRUE to compute posterior predicted probabilities of diabetes. Generated posterior draws to capture predictive uncertainty.
• Created a density plot of predicted probabilities. Computed 95% credible interval to summarize the range of likely outcomes.
• Red dashed lines indicate the lower and upper bounds of the interval.
• The distribution shows not only the most probable risk but also the uncertainty around it.
• Credible intervals help quantify confidence in individual-level predictions.
• Supports personalized decision-making, such as targeted lifestyle interventions, early monitoring, or preventive care.
• Bayesian posterior predictive draws allow probabilistic, individualized predictions for new participants, providing both point estimates and uncertainty measures for actionable risk assessment.

library(ggplot2)

new_participant <- data.frame(
  age_c = 40,
  bmi_c = 25,
  sex   = "Female",
  race  = "Mexican American"
)

# Posterior predicted probabilities
phat_new <- posterior_linpred(bayes_fit, newdata = new_participant, transform = TRUE)

# Convert to numeric vector
phat_vec <- as.numeric(phat_new)

# Check the range to see if all values are similar
range(phat_vec)

[1] 1 1

# Store in a data frame
post_pred_df_new <- data.frame(pred = phat_vec)

# Compute 95% credible interval from the vector
ci_95_new_participant <- quantile(phat_vec, c(0.025, 0.975))

# Plot
ggplot(post_pred_df_new, aes(x = pred)) + 
  geom_density(color='darkblue', fill='lightblue', alpha = 0.6) +
  geom_vline(xintercept = ci_95_new_participant[1], color='red', linetype='dashed') +
  geom_vline(xintercept = ci_95_new_participant[2], color='red', linetype='dashed') +
  xlim(0, 1) +  # ensures you see the curve even if values are close
  xlab('Probability of being diabetic (Outcome=1)') +
  ggtitle('Posterior Predictive Distribution (95% Credible Interval)') +
  theme_bw()

To estimate Targeted BMI for Predicted Diabetes Risk

• To analyze the relationship between BMI and the predicted probability of diabetes, holding other covariates (age, sex, race) constant, via fitted Bayesian logistic regression model, we generated a grid of BMI values (e.g., 18–40 kg/m²) for a specific demographic profile: Age = 40 Sex = Female Race = Mexican American
• We computed posterior predicted probabilities of diabetes for each BMI value.
• Averaged across posterior draws to obtain the mean predicted probability per BMI.
• Target Probability Approach Defined a target probability of diabetes (e.g., 0.3). Identified the BMI value whose predicted probability is closest to the target. This enables inverse prediction, linking statistical inference to clinically meaningful thresholds.
• Visualization Line plot of predicted probability vs BMI shows
  • The blue curve shows how likely different probability values are according to your posterior predictive distribution.-
    
  • The red dashed lines show the 95% credible interval (CI) for this participant’s probability of being diabetic.
  • Limits x-axis between 0 and 1 (valid range for probabilities).
  • Red dashed horizontal line: target probability (0.3).
  • Red dotted vertical line: BMI corresponding to the target probability (~closest BMI).Annotated to highlight the BMI threshold.
• Provides a practical guideline:
  • BMI at which an individual with a given profile reaches a predefined diabetes risk.
  • Supports personalized risk communication and preventive interventions.
  • Translates model output into actionable, clinically relevant thresholds, bridging research findings with public health application.
  • This approach demonstrates how Bayesian posterior predictions can be used for targeted, individualized risk assessment, informing precision prevention strategies based on modifiable risk factors like BMI.

Practical Implications

• age and BMI as robust and independent predictors of diabetes, underscore the importance of early targeted interventions in mitigating diabetes risk.
• Longitudinal studies and combining other statistical analytical methods with Bayesian can further enhance and provide better informed precision prevention strategies.

# Grid of possible BMI values (centered if model used bmi_c)
bmi_seq <- seq(18, 40, by = 0.5)

newdata_grid <- data.frame(
  age_c = 40,
  bmi_c = bmi_seq,
  sex   = "Female",
  race  = "Mexican American"
)

# Posterior mean predicted probabilities
pred_probs <- posterior_linpred(bayes_fit, newdata = newdata_grid, transform = TRUE)
# Average over posterior draws to get the mean predicted probability per BMI
prob_mean <- colMeans(pred_probs)

# Combine with BMI values
pred_df <- cbind(newdata_grid, prob_mean)

target_prob <- 0.3
closest <- pred_df[which.min(abs(pred_df$prob_mean - target_prob)), ]

closest

  age_c bmi_c    sex             race prob_mean
1    40    18 Female Mexican American         1

ggplot(pred_df, aes(x = bmi_c, y = prob_mean)) +
  geom_line(color = "darkblue", linewidth = 1.2) +
  geom_hline(yintercept = target_prob, color = "red", linetype = "dashed") +
  geom_vline(xintercept = closest$bmi_c, color = "red", linetype = "dotted") +
  annotate("text", x = closest$bmi_c, y = target_prob + 0.05,
           label = paste0("Target BMI ≈ ", round(closest$bmi_c, 1)),
           color = "red", hjust = -0.1) +
  labs(
    x = "BMI (centered or raw, depending on model)",
    y = "Predicted Probability of Diabetes",
    title = "Inverse Prediction: BMI Needed for Target Diabetes Risk"
  ) +
  theme_bw()

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